Quadrature-based Moment Methods Research Articles

Uncertainty quantification and sensitivity analysis of volcanic columns models: Results from the integral model PLUME-MoM

The behavior of plumes associated with explosive volcanic eruptions is complex and dependent on eruptive source parameters (e.g. exit velocity, gas fraction, temperature and grain-size distribution). It is also well known that the atmospheric environment interacts with volcanic plumes produced by explosive eruptions in a number of ways. The wind field can bend the plume but also affect atmospheric air entrainment into the column, enhancing its buoyancy and in some cases, preventing column collapse. In recent years, several numerical simulation tools and observational systems have investigated the action of eruption parameters and wind field on volcanic column height and column trajectory, revealing an important influence of these variables on plume behavior. In this study, we assess these dependencies using the integral model PLUME-MoM, whereby the continuous polydispersity of pyroclastic particles is described using a quadrature-based moment method, an innovative approach in volcanology well-suited for the description of the multiphase nature of magmatic mixtures. Application of formalized uncertainty quantification and sensitivity analysis techniques enables statistical exploration of the model, providing information on the extent to which uncertainty in the input or model parameters propagates to model output uncertainty. In particular, in the framework of the IAVCEI Commission on tephra hazard modeling inter-comparison study, PLUME-MoM is used to investigate the parameters exerting a major control on plume height, applying it to a weak plume scenario based on 26 January 2011 Shinmoe-dake eruptive conditions and a strong plume scenario based on the climatic phase of the 15 June 1991 Pinatubo eruption.

Limitations of simple mass transfer models in polydisperse liquid–liquid dispersions

In this work the limitations of standard approaches, commonly used to describe mass transfer in fluid–fluid systems are critically discussed. Emphasis is placed on polydisperse systems, such as liquid–liquid dispersions. In the simplest possible approach the fluid–fluid system is assumed to be monodisperse: namely all the elements of the disperse phase have the same properties. This very simple (albeit inaccurate) approach is the one typically employed for design, scale-up and optimization of unit operations and of the corresponding equipments. The adequacy (or inadequacy) of this model depends on the competition between three phenomena: droplet coalescence, breakage and mass transfer (often enhanced by chemical reactions). When coalescence and breakage dominate, the only polydispersity that has to be accounted for is droplet size, whereas on the contrary, when mass transfer dominates the process, very often polydispersity with respect to both size and chemical composition must be considered. This can be efficiently done by solving the so-called population balance equation with, for example, quadrature-based moment methods. In this work a strategy to evaluate a-priori the complexity of the model to be used, based on time scales, is proposed and tested.

An extended quadrature-based moment method with log-normal kernel density functions

An extended quadrature method of moments (EQMOM) with log-normal kernel density functions is developed in this work, and applied to the solution of a population balance equation (PBE) for aggregation and breakup, coalescence, and condensation problems. The cases with one and two kernel density functions are studied analytically, and the existence of an analytical solution is shown. A numerical procedure based on the work of Yuan et al. (2012) is adopted to address cases with a larger number of kernel density functions. Results for the reconstructed number density function (NDF), the time evolution of the zero-order moment and of the mean particle size are compared with those obtained from the rigorous solution of the PBE reported by Vanni (2000) for the cases of aggregation and breakup. A problem concerning coalescence and one regarding condensation, both with analytical solution, are also examined. The results obtained with the proposed approach are compared to those provided by EQMOM with gamma kernel densities. Satisfactory results were obtained for the reconstructed distribution. Excellent agreement was observed between the rigorous solution and the approximated one for the time evolution of the total number density and the mean particle size.

An extended quadrature‐based mass‐velocity moment model for polydisperse bubbly flows

AbstractAccurately predicting polydisperse bubbly flow is a nontrivial task due to the complexity of the bubble number density function (NDF) and the strong dependence of the instantaneous bubble velocity on the bubble size and shape. To describe polydisperse bubbly flow, a joint mass‐velocity NDF is adopted in this work. In the absence of mass transfer between phases and coalescence or breakage, the bubble mass is a conserved quantity from which the bubble size and shape can be found given the liquid pressure and surface tension. Quadrature‐based moment methods (QBMM) are applied to solve numerically the kinetic equation of the joint NDF using the extended quadrature method of moments (EQMOM) coupled with an open‐source incompressible Navier–Stokes solver for the liquid phase. Transport equations for the joint mass‐velocity moments are derived from a kinetic equation for the joint NDF and closure is attained using a monokinetic NDF valid in the limit of small bubble Stokes number. The integer moments with respect to mass are used to reconstruct the continuous univariate NDF with EQMOM, while the joint mass‐velocity moments are used to determine the bubble velocity as a continuous function of the bubble mass. The model is first applied to simulate a quasi‐2‐D bubble column with different aeration profiles and a narrow bubble size distribution in order to validate the approach with experimental data from the literature. Additional cases with a wide continuous bubble size distribution are used to show the ability of the modelling approach to describe polydisperse bubbly flows.

Modeling and simulation of turbulent polydisperse gas-liquid systems via the generalized population balance equation

This article reviews the most critical issues in the simulation of turbulent polydisperse gas-liquid systems. Here the discussion is limited to bubbly flows, where the gas appears in the form of separate individual bubbles. First, the governing equations are presented with particular focus on the generalized population balance equation (GPBE). Then, the mesoscale models defining the evolution of the gas-liquid system (e.g., interface forces, mass transfer, coalescence, and breakup) are introduced and critically discussed. Particular attention is devoted to the choice of the drag model to properly simulate dense gas-liquid systems in the presence of microscale turbulence. Finally, the different solution methods, namely, Lagrangian and Eulerian, are presented and discussed. The link between mixture, two- and multi-fluid models, and the GPBE is also analyzed. Eventually, the different methodologies to account for polydispersity, with focus on Lagrangian or direct simulation Monte Carlo methods and Eulerian quadrature-based moment methods, are also presented. A number of practical examples are discussed and the review is concluded by presenting the advantages and disadvantages of the different methods and the corresponding computational costs.

Realizable high-order finite-volume schemes for quadrature-based moment methods applied to diffusion population balance equations

Population balance equations with advection and diffusion terms can be solved using quadrature-based moment methods. Recently, high-order realizable finite-volume schemes with appropriate realizabi...

Numerical aspects of direct quadrature-based moment methods for solving the population balance equation

Direct-quadrature generalized moment based methods were analysed in terms of accuracy, computational cost and robustness for the solution of the population balance problems in the [0,∞) and [0,1] domains. The minimum condition number of the coefficient matrix of their linear system of equations was obtained by global optimization. An heuristic scaling rule from the literature was also evaluated. The results indicate that the methods based on Legendre generalized moments are the most robust for the finite domain problems, while the DQMoM formulation that solves for the abscissas and weights using the heuristic scaling rule is the best for the infinite domain problems.

Realizable high-order finite-volume schemes for quadrature-based moment methods applied to diffusion population balance equations

Population balance equations with advection and diffusion terms can be solved using quadrature-based moment methods. Recently, high-order realizable finite-volume schemes with appropriate realizability criteria have been derived for the advection term. However, hitherto no work has been reported with respect to realizability problems for the diffusion term. The current work focuses on developing high-order realizable finite-volume schemes for diffusion. The pitfalls of existing finite-volume schemes for the diffusion term based on the reconstruction of moments are discussed, and it is shown that realizability can be guaranteed only with the 2nd-order scheme and that the realizability criterion for the 2nd-order scheme is the same as the stability criterion. However, realizability of moments cannot be guaranteed when higher-order moment-based reconstruction schemes are used. To overcome this problem, realizable high-order finite-volume schemes based on the reconstruction of weights and abscissas are proposed and suitable realizability criteria are derived. The realizable schemes can achieve higher than 2nd-order accuracy for problems with smoothly varying abscissas. In the worst-case scenario of highly nonlinear abscissas, the realizable schemes are 2nd-order accurate but have lower error magnitudes compared to existing schemes. The results obtained using the realizable high-order schemes are shown to be consistent with those obtained using the 2nd-order moment-based reconstruction scheme.

Equilibrium-Eulerian LES Model for Turbulent Poly-dispersed Particle-laden Flow

Abstract An efficient Eulerian method for poly-dispersed particles in turbulent flows is implemented, verified and validated for a channel flow. The approach couples a mixture model with a quadrature-based moment method for the particle size distribution in a LES framework, augmented by an approximate deconvolution method to reconstructs the unfiltered velocity. The particle velocity conditioned on particle size is calculated with an equilibrium model, valid for low Stokes numbers. A population balance equation is solved with the direct quadrature method of moments, that efficiently represents the continuous particle size distribution. In this first study particulate processes are not considered and the capability of the model to properly describe particle transport is investigated for a turbulent channel flow. First, single-phase LES are validated through comparison with DNS. Then predictions for the two-phase system, with particles characterised by Stokes numbers ranging from 0.2 to 5, are compared with Lagrangian DNS in terms of particle velocity and accumulation at the walls. Since this phenomenon (turbophoresis) is driven by turbulent fluctuations and depends strongly on the particle Stokes number, the approximation of the particle size distribution, the choice of the sub-grid scale model and the use of an approximate deconvolution method are important to obtain good results. Our method can be considered as a fast and efficient alternative to classical Lagrangian methods or Eulerian multi-fluid models in which poly-dispersity is usually neglected.

Simulation of Mono- and Bidisperse Gas-Particle Flow in a Riser with a Third-Order Quadrature-Based Moment Method

Gas-particle flows can be described by a kinetic equation for the particle phase coupled with the Navier–Stokes equations for the fluid phase through a momentum exchange term. The direct solution of the kinetic equation is prohibitive for most applications due to the high dimensionality of the space of independent variables. A viable alternative is represented by moment methods, where moments of the velocity distribution function are transported in space and time. In this work, a fully coupled third-order, quadrature-based moment method is applied to the simulation of mono- and bidisperse gas-particle flows in the riser of a circulating fluidized bed. Gaussian quadrature formulas are used to model the unclosed terms in the moment transport equations. A Bhatnagar–Gross–Krook (BGK) collision model is used in the monodisperse case, while the full Boltzmann integral is adopted in the bidisperse case. The predicted values of mean local phase velocities, rms velocities, and particle volume fractions are compared with the Euler–Lagrange simulations and experimental data from the literature. The local values of the time-average Stokes, Mach, and Knudsen numbers predicted by the simulation are reported and analyzed to justify the adoption of high-order moment methods as opposed to models based on hydrodynamic closures.

Development of a hybrid sectional quadrature-based moment method for solving population balance equations

Various ad hoc methods have been proposed for solving population balance equations (PBEs). Relatively little discussion has taken place to generalize these discretization approaches. In this paper, a flexible method is presented for the discrete solution of PBEs. The method is a generalization of similar techniques. The approach allows for complete control of the discretization and order of approximation of the solution. A simple error analysis of the method is provided. Simulations were performed for pure random breakage, constant kernel aggregation, additive kernel aggregation, combined random breakage/constant kernel aggregation, and free-molecular Brownian coagulation. The numerical results are compared to exact solutions and simulated results from the literature. It was found that increasing the quadrature order within the sections is a more efficient means to increase accuracy as opposed to finer discretization.

An extended quadrature method of moments for population balance equations

Population balance equations (PBE) for a number density function (NDF) arise in many applications of aerosol technology. Thus, there has been considerable interest in the development of numerical methods to find solutions to PBE, especially in the context of spatially inhomogeneous systems where moment realizability becomes a significant issue. Quadrature-based moment methods (QBMM) are an important class of methods for which the accuracy of the solution can be improved in a controlled manner by increasing the number of quadrature nodes. However, when a large number of nodes is required to achieve the desired accuracy, the moment-inversion problem can become ill-conditioned. Moreover, oftentimes pointwise values of the NDF are required, but are unavailable with existing QBMM. In this work, a new generation of QBMM is introduced that provides an explicit form for the NDF. This extended quadrature method of moments (EQMOM) approximates the NDF by a sum of non-negative weight functions, which allows unclosed source terms to be computed with great accuracy by increasing the number of quadrature nodes independent of the number of transported moments. Here, we use EQMOM to solve a spatially homogeneous PBE with aggregation, breakage, condensation, and evaporation terms, and compare the results with analytical solutions whenever possible. However, by employing realizable finite-volume methods, the extension of EQMOM to spatially inhomogeneous systems is straightforward.

Conditional quadrature method of moments for kinetic equations

Kinetic equations arise in a wide variety of physical systems and efficient numerical methods are needed for their solution. Moment methods are an important class of approximate models derived from kinetic equations, but require closure to truncate the moment set. In quadrature-based moment methods (QBMM), closure is achieved by inverting a finite set of moments to reconstruct a point distribution from which all unclosed moments (e.g. spatial fluxes) can be related to the finite moment set. In this work, a novel moment-inversion algorithm, based on 1-D adaptive quadrature of conditional velocity moments, is introduced and shown to always yield realizable distribution functions (i.e. non-negative quadrature weights). This conditional quadrature method of moments (CQMOM) can be used to compute exact N-point quadratures for multi-valued solutions (also known as the multi-variate truncated moment problem), and provides optimal approximations of continuous distributions. In order to control numerical errors arising in volume averaging and spatial transport, an adaptive 1-D quadrature algorithm is formulated for use with CQMOM. The use of adaptive CQMOM in the context of QBMM for the solution of kinetic equations is illustrated by applying it to problems involving particle trajectory crossing (i.e. collision-less systems), elastic and inelastic particle–particle collisions, and external forces (i.e. fluid drag).

Limitations of quadrature-based moment methods for modeling inhomogeneous polydisperse fluidized powders

Computational fluid dynamics (CFD) is extensively used to investigate the behavior of dense fluidized suspensions. Often modelers assume that these are formed by few solid phases of particles with constant size. But real powders are continuously distributed over the particle size, and their distribution functions change continuously in time and space reflecting the physical and chemical phenomena occurring within the system. To account for this key feature, models have to include a population balance equation (PBE), which needs to be solved in place of or along with the customary fluid dynamic transport equations. The recently developed quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) permit to solve PBEs in commercial CFD codes at relatively low computational cost. These methods, however, still need testing in the context of multiphase flows. Investigating a simple problem, namely the dynamics of two inert polydisperse fluidized suspensions initially segregated, we highlight an important limitation of these methods, which fail to properly model diffusion in real space. We explain where the problem originates and comment on a possible way to overcome it. To conclude the work, we discuss some simulations based on the original and revised formulations of the methods, describing how the code numerics affects the results.

Realizable high-order finite-volume schemes for quadrature-based moment methods

Dilute gas–particle flows can be described by a kinetic equation containing terms for spatial transport, gravity, fluid drag and particle–particle collisions. However, direct numerical solution of kinetic equations is often infeasible because of the large number of independent variables. An alternative is to reformulate the problem in terms of the moments of the velocity distribution. Recently, a quadrature-based moment method was derived for approximating solutions to kinetic equations. The success of the new method is based on a moment-inversion algorithm that is used to calculate non-negative weights and abscissas from the moments. The moment-inversion algorithm does not work if the moments are non-realizable, which might lead to negative weights. It has been recently shown [14] that realizability is guaranteed only with the 1st-order finite-volume scheme that has an inherent problem of excessive numerical diffusion. The use of high-order finite-volume schemes may lead to non-realizable moments. In the present work, realizability of the finite-volume schemes in both space and time is discussed for the 1st time. A generalized idea for developing realizable high-order finite-volume schemes for quadrature-based moment methods is presented. These finite-volume schemes give remarkable improvement in the solutions for a certain class of problems. It is also shown that the standard Runge–Kutta time-integration schemes do not guarantee realizability. However, realizability can be guaranteed if strong stability-preserving (SSP) Runge–Kutta schemes are used. Numerical results are presented on both Cartesian and triangular meshes.

Modeling of bubble-column flows with quadrature-based moment methods

Bubble-column reactors are frequently employed in the biological, chemical and petrochemical industries. This paper presents a novel approach to model bubble-column flows using quadrature-based moment methods (QBMM). A fully two-way coupled flow solver is developed that solves the incompressible Navier-Stokes equation for the liquid phase and moment transport equations for the dispersed bubble phase. The moment transport equations for the dispersed bubble phase are solved using a kinetic theory approach. Contributions from the liquid-phase pressure gradient, vorticity, drag, virtual mass and gravity are accounted for in the bubble-phase force balance. The solution algorithm and coupling procedure are described in detail, and results are presented for a 2-D bubble column with two different gas flow rates (1.6 and 8.0 l/min).

Advanced continuum modelling of gas-particle flows beyond the hydrodynamic limit

The accurate prediction of dilute gas-particle flows using Euler–Euler models is challenging because particle–particle collisions are usually not dominant in such flows. In other words, in dilute flows the particle Knudsen number is not small enough to justify a Chapman–Enskog expansion about the collision-dominated near-equilibrium limit. Moreover, due to the fluid drag and inelastic collisions, the granular temperature in gas-particle flows is often small compared to the mean particle kinetic energy, implying that the particle-phase Mach number can be very large. In analogy to rarefied gas flows, it is thus not surprising that two-fluid models fail for gas-particle flows with moderate Knudsen and Mach numbers. In this work, a third-order quadrature-based moment method, valid for arbitrary Knudsen number, coupled with a fluid solver has been applied to simulate dilute gas-particle flow in a vertical channel with particle-phase volume fractions between 0.0001 and 0.01. In order to isolate the instabilities that arise due to fluid-particle coupling, a fluid mass flow rate that ensures that turbulence would not develop in a single phase flow (Re = 1380) is employed. Results are compared with the predictions of a two-fluid model with standard kinetic theory based closures for the particle phase. The effect of the particle-phase volume fraction on flow instabilities leading to particle segregation is investigated, and differences with respect to the two-fluid model predictions are examined. The influence of the discretization on the solution of both models is investigated using three different grid resolutions. Radial profiles of phase velocities and particle concentration are shown for the case with an average particle volume fraction of 0.01, showing the flow is in the core-annular regime.

Eulerian Quadrature-Based Moment Models for Dilute Polydisperse Evaporating Sprays

Dilute liquid sprays can be modeled at the mesoscale using a kinetic equation, namely the Williams–Boltzmann equation, containing terms for spatial transport, evaporation and fluid drag. The most common method for simulating the Williams–Boltzmann equation uses Lagrangian particle tracking wherein a finite ensemble of numerical “parcels” provides a statistical estimate of the joint surface area, velocity number density function (NDF). An alternative approach is to discretize the NDF into droplet size intervals, called sections, and to neglect velocity fluctuations conditioned on droplet size, resulting in an Eulerian multi-fluid model. In comparison to Lagrangian particle tracking, multi-fluid models contain no statistical error (due to the finite number of parcels) but they cannot reproduce the particle trajectory crossings observed in Lagrangian simulations of non-collisional kinetic equations. Here, in order to overcome this limitation, a quadrature-based moment method is used to describe the velocity moments. When coupled with the sectional description of droplet sizes, the resulting Eulerian multi-fluid, multi-velocity model is shown to capture accurately both particle trajectory crossings and the size-dependent dynamics of evaporation and fluid drag. Model validation is carried out using direct comparisons between the Lagrangian and Eulerian models for an unsteady free-jet configuration with mono- and polydisperse droplets with and without evaporation. Comparisons between the Eulerian and Lagrangian instantaneous number density and gas-phase fuel mass fraction fields show excellent agreement, suggesting that the multi-fluid, multi-velocity model is well suited for describing spray combustion.

A fully coupled quadrature-based moment method for dilute to moderately dilute fluid–particle flows

A third-order quadrature-based moment method for simulating dilute and moderately dilute fluid–particle flows has been implemented with full coupling in a computational fluid dynamics code. The solution algorithm for the particle phase uses a kinetic-based finite-volume technique to solve the velocity moment equations derived from kinetic theory. The procedure to couple the particle-phase volume-fraction and momentum equations with the Eulerian solver for the fluid phase is explained in detail. As an example application, simulations of a particle-laden vertical channel flow at fluid-phase Reynolds number 1379 and particle Stokes numbers 0.061 and 0.61 were carried out. The fluid and particle velocities, particle-phase volume fraction and granular temperature were observed to reach a steady state in the case of Stokes number 0.061, while instabilities that led to the formation of structures and initiated the particle segregation process were observed in the case with the higher Stokes number. These results are validated against results from a classical two-fluid model derived from the kinetic theory of granular flows in the small Knudsen number limit, and Euler–Lagrange simulations of the same flow.

Higher-order quadrature-based moment methods for kinetic equations

Kinetic equations containing terms for spatial transport, body forces, and particle–particle collisions occur in many applications (e.g., rarefied gases, dilute granular gases, fluid-particle flows). The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. A useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function. Closure of the moment equations is challenging for flows sufficiently far away from the Maxwellian limit. In previous work, a quadrature-based third-order moment closure was derived for approximating solutions to the kinetic equation for arbitrary Knudsen number. A key component of quadrature-based closures is the moment-inversion algorithm used to find the non-negative weights and velocity abscissas. Here, a robust inversion procedure is proposed for three-component velocity moments up to ninth order. By reconstructing the velocity distribution function, the spatial fluxes in the moment equations are treated using a kinetic-based finite-volume solver. Because the quadrature-based moment method employs the moment transport equations directly instead of a discretized form of the kinetic equation, the mass, momentum and energy are conserved for arbitrary Knudsen and Mach numbers. The computational algorithm is tested for the Riemann shock problem and, for increasing Knudsen numbers (i.e. larger deviations from the Maxwellian limit), the accuracy of the moment closure is shown to be determined by the discrete representation of the spatial fluxes.